On the density of some wiener functionals : an application of Malliavin calculus

Publicacions Ma emá iques, Vol 36 (1992), 981-987 . A bs ac ON THE DENSITYOF SOME WIENER FUNCTIONALS : AN APPLICATION OF MALLIAVIN CALCULUS ANTON1 SINTEs BLANC Dedica ed o P o esso Pe e Menal i B u al, in memo iam Using a ep esen a ion as an in ini e linea combina ion o chi- squa e independen andom a iables, i is shown ha some Wiene unc ionals, appea ing in empi ical cha ac e is ic p ocess asymp o ic heo y, ha e densi ies which a e empe ed in he p op- e ly in ini e case and exponen ially decaying in he ini e case . In oduc ion Le F be a p obabili y dis ibu ion unc ion on R, wi h cha ac e is- ic unc ion c( ) := E(exp{i X}), X being a eal andom a iable wi h dis ibu ion unc ion F . Le X 1 , X2, ... be a sequence o independen copies o X de ined on some p obabili y space (S2, .P, P), and F n he empi ical dis ibu ion unc- ion o he i s n a iables . The empi ical cha ac e is ic unc ion c ,( ) is he ( andom) cha ac e - is ic unc ion o F, and has been used in se e al s a is ical applica ions since a leas C ame 's amous book . In he la e 70's a sys ema ic s udy o i s p ope ies and s a is ical ap- plica ions was ini ia ed by Feue e ge and Mu eika . In pa icula hey p o ed he i s limi heo ems o he empi ical cha ac e is ic p ocess Y,~( ) := ~,l,-n,(c  ( )-c( )), unde s ong mo ien condi ions on F, see [2] . In 1981 M . Ma cus, [3], ound necessa y and su icien condi ions o he p ocesses {X,,( )},°,°__i, E o weakly con e ge o a limi p ocess o he ollowing o m + Y( ) :=  +00 exp{i x}d 00 I : 982  A . SINTES BLANC whe e B is B ownian b idge on [0,1], as C[T1,T2] alued andom ele- men s . The se me yea S . Csó gó, [1, Theo em 3], .ob ained s ong app oxima- ions o Yn,( ), unde he assump ion h(x)[1 - F(x) +F(- .x)] = O(1) ,  as x goes o +oo whe e h is a con inuous unc ion on (0, +oo) and h(x)x - " inc eases o +oo as x inc eases o +oo, o some posi i o u . F om his s ong app oxima ions S . Csó gó de i ed a es . ,o con e - gence o se e al unc iónals o he pa hs o he empi ical cha ac e is ic p ocese, unde he u he assump ion o exis en e and boundedness o a densi y o he limi unc ional . He e we wan o s udy he ollowing one which is use ul in es ing o symme y o F ., namely :=  ¡'Tz . [I~ n(X( ))] 2dH( ) whe e 0-V is a gi en dis ibu ion unc ion wi h suppo in [T 1 ,T2], and -oo < T I < T2 < +oo . The main esul in he p esen wo k is ha wi h g ea gene ali y, T has a densi y which in ac sa is ies much s onge boundedness condi ions han hose needed o Csó gó's a es o con e gen e o hold ue . I is he ollowing Theo em . Assume ha he andom a iable T, as de ined abone, is non degene a e ( his is a condi ion on bo h, F and Hl) . Then T has a smoo h densi y, which is ei he a empe ed unc ion o an exponen ially decaying one (a in ini y) . The main ing edien e o he p oo a e Mallia in calculus, Cauchy's o mula and Fou ie ans o m . In he nex pa ag aph we ecall, e y b ie ly, he p incipal de ini ions and esul s we a e going o use, and gi e some e e en es whe e comple e and de ailed exposi ions can be ound . Finally, in pa ag aph, 3 we gi e he p oo o he heo em . The ools Le L2 [0, 1] be he Hilbe space o squa e in eg able unc ions wi h espec o he Lebesgue measu e on he Bo el u- ield, C3, i 1 he uni in e al [0,1] . Fo hE L2 [0,1] ; deno e VY(h) he Wiene in eg al o h wi h espec o he Wiene p ocess, W, on [0,1] . We hink o W as a Gaussian o hogonal mensu e on he space ([0, 1], L3, A), i is, a ze o mean Gaussian p ocess {W (B) : B E 13} de ined on some p obabili y space (S2, F ; P), wi h co- a iance unc ion gi en by E(W(BI^B2)) = A(B 1 n B2), whe e A is Lebesgue measu e on [0,1] . A ound 1950 K . I o showed ha each squa e in eg able unc ional F E  P) can be de eloped in he o m whe e I,,(  j a e mul iple I o-Wiene in eg als o he (de e minis ic) unc ions  , E L2([0, 1]m, 13' n , A' n ) . In pa icula II ( I) = W( 1) . This expansion is some imes known as he Wiene chaos decomposi ion o F . S oock's o mula iden i ies he ke nels, , in e ms o i e a ed Malli- a in de i a i es o F : Fo F E L2 (52, .F, P) and hE L2 [0,1], he Mallia in de i a i e o F in he h di ec ion, Dj L F, can be de ined as i he se ies con e ges in L2 (S2, .F, P), whe e (, ) is inne p oduc .  All his (and much mo e) can be ound in [4] and [5] . To apply his heo y o ou unc ional T, we w i e i down in e ms o as ollows : AN APPLICATION OF MALLIAVIN CALCULUS  983 D h F := 7n=1 00 F= E (F) + Y~ I ( m ) =1 ,,,, = (m!)-1E(D-F) . (h  I a-1( a( l . . . , a-1,*))i T= TZ [W(h )] 2 dH( ) T l whe e h (y) := sin( F -1 (y)) - Im(c( )), o y E [0,1], c( ) being he cha ac e is ic unc ion o F, Le . c( ) := E(exp{iA}), whe e k is a F- dis ibu ed andom a iable . Le us calcula e he Mallia in de i a i es D[W (h )] 2 = 2W(h )h D 2 [W(h )] 2 = 2h ® h and he de i a i es o o de g ea e o equal han 3 a e all ze o . 984  A . SINTEs BLANC Hence ; by linea i y and om S oock's o mula we ge . 1 T' D 2 T _  2h ®h d~l ( ) T 1 7T = E(T) + 12 ( .l  2 T, h ® h dH( ) Now we ecall some mo e de ini ions and esul s ha will also be needed in he nex pa ag aph . A unc ion on R is called a empe ed unc ion i i is a smoo h unc ion and o each N E Nl ( he se o na u al numbe s) sup sup(1+x 2 ) N j (p )(x)1 < + oo p<N xER whe e (p) is he p- h de i a i e o . Le S deno e he space o empe ed unc ions on R . The Fou ie ans o m o a unc ion E S is de ined as ( ) :_ (27 ) - z ~  exp(-i x) (x)dx 00 I u ns ou ha his ans o ma ion de ines a con inuous one- o-one mapping o S on o S, whose in e se is also con inuous . Mo eo e , i has pe iod 4, due o he in e sion o mula which is undamen al +0 ( .x) = (27 ) - z  exp(i x) ( )d . _ 00 A e y good e e en e o his is [6] . P oo o he heo em The idea o he p oo is qui e simple : o show ha he cha ac e is ic unc ion o T is a empe ed unc ion in he p ope ly in ini e case ; and di ec calcula ion in he ini e case (dis ine ion o he wo cases will be clea in a momen ) . Le us go back o S oock's ep esen a ion o T ; and ecall he explici o m o he ke nel in he' double I o-Wiene in eg al he e . AN APPLICATION O 1VIALLIAVIN CALCULUS  985 I is known ha such a ke nel can be de eloped in a L 2 -con e gen se ies . K( ,s) = L Aiei( )ei(s) whe e A ;,, ( espec i ely e?(-)), a e he eigen alues, ( espec i ely eigen ec- o s) o he ollowing sel -adjoin non-nega i e in eg al ope a o I o( .)- K(-, s)O(s)ds 0 en L 2 [0 ; 1] ; he e&)'s can be assumed o be a comple e o hono mal sys em in L 2 [0, 1], and j :° °I A ;, < +oo . I is also well known ha o hE L 2 [0, 1], wi h 1111112 = 1 I2(h (D h) = W(h) 2 - 1 and combina ion o his ac s leads us o he ep esen a ion we had in mind, namely _ E(T) +  ~i(W(ei)2 - 1) _  AáW(ei)2 wi h F-00, ' i <+ 00 . We ema k he e ha om his ep esen a ion i is clea ha T is non-degene a e i S oock's ke nel, D2 T, is non-degene a e . To p oceed we dis inguish wo cases : he one whe e only a ini e nu i- be o he Ai a e non ze o, and he one whe e he e a e in ini ely many non ze o . We ea only he second case, as he i s one is e y elemen- a y . So, o inish he p oo o he heo em i is enough o p o e nex lemma, Lemma . Le {%i}.°°, be independen iden ically dis ibu ed andom a iables wi h a common chi-squa e dis ibu ion, ¡ .e . a dis ibu ion wi h densi yx - z -exp(-2)110,-1( ,) . Le {Ai }°° I be a noninc easing sequence o posi i e eal numbe s such ha  ñ i < +oo . Then he andom a iable ,1  AiXi has a empe ed p obabili y densi y anc ion . P oo - I is well known ha he cha ac e is ic unc ion o he andom a iable .D is 00 . ( ) _ l (1 - 2A,,i )  2 k-I 986  A . SINTEs BLANC I is easily seen ha his unc ion possesses an analy ic con inua ion on he s ip - ( 4, 1) -1 < Im(z) < ( 4, 1) -1 , which is o he same o m, Le . (z) = l (1 - 2Akiz) -12 k=1 whe e i := V /- -- 1 . Obse e ha o zin his s ip Z < Re(1 - 2¡, z) < 2 , so ha we can use he b anch o y /'z - which coincides wi h he usual squa e oo unc ion on he posi i e eal numbe s ., and we can apply Theo em (15 .6) om [7] ; (he e we use he hypo hesis -°°1 ~i <+ oo) . Fo > 0 apply Cauchy's o mula o he ec angle F de ined by he poin s ( /2) T- i(4A1) -1 , 2 :F i(4A1) - ' ; o ge (P)( ) _ (P!) 1 : 1  (z - ( z ) ) P+1 dz j=1 j whe e {Fj}~_1 a e he ou sides o he ec angle F . To bound his in eg als we use he ac ha o each na u al numbe MEN Fo ins an e he in eg al on he le side, le us say F I , o he ec angle F, can be bounded as ollows (4A1) -1 (( /2) + is)j j(( /2) - is)j -1- Pds < (4ñ1) -1 M  1 < 2PAi 1 P +1  (( 1 /4) + A2 2)l -` k=1 whe e we used (1) ; and he same ype o bound is ob ained o he in eg al on he igh side, 1'3, o F . In a simila way he bound in (2) is used o ea he in eg als on he ho i zon al sides o he ec angle, F2 and I' 4 . And his p o es he lemma in he case > 0 . (1) m _, a SUP j « /2) + is)~ < (( 1 / 4 ) + isi<(4a1)-1 (~ ~kk 2)) k=1 ( 2 ) n~ 1 _ SUP l (s + i(4a1) -1 )I ~ (~ (1 + >,2 2» ( /2)<isi<2 k=1 AN APPLICATION 01 " MALLIAVINCALCULUS  987 Fo < 0, wi h small changos, all wo ks he same way, and o = 0 he e is no p oblem a all . Thus he lemma is p o ed . Rema k . In he abo e p oo we could ha e used he asymp o ic ep- esen a ion ob ained in [8], ins ead o ou lemma . I is clea om [8] ha he densi y (x) is expo ien ially decaying a in ini y, e en in he p ope ly in ini o case . Howe e , i is no clea how can we ge he em- pe a e cha ac e o om Zolo a e 's ep esen a ion, wi hou u he wo k . In ha poin ou p oo seems o be sho e and clea e . Acknowledgmen s . 1 wan o hank P o esso s Paul Mallia in and Da id Nuala o hei kind a en ion and help on some p oblems ela ed o he subjec o his pape . Thanks also o P o esso E a is Giné, o poin me ou e e en e [8] . Re e en es 1 .  S . Csói3 .Gó, Limi beha io o he empi ical cha ac e is ic unc ion, The Annals o P obabili y 9, no . 1 (1981), 130-144 . 2 .  A . FEUERVERGER AND R . A . MUREIKA, The empi ical cha ac e - is ic unc ion and i s applica ions, Ann . S a is . 5 (1977), 88-97 . 3 .  M . B . MARCUS, Weak con e gen e o he empi ical cha ac e is ic unc ion, Au a . P obabili y 9 (1981), 194-201 . 4 . D . NUALART AND M . ZAKAI, Gene alized s ochas ic in eg als and he Mallia in calculus, P obab . Theo y Rel . Fields 73 (1986), 255-280 . 5 .  D . NUALART AND M . ZAKAI, Gene alized mul iple s ochas ic in e- g als and he ep esen a ion o Wiene unc ionals, S ochas ics 23 (1988),311-330 . 6 .  W . RUDIN, "Func ional Analysis," McG aw-Hill, 1973 . 7 .  W . RUDIN, "Real and complex analysis," McG aw-Hill, 1970 . 8 .  V . M . ZOLOTAREV, Conce ning a ce ain p obabili y p oblem, The- o y o P obab . Appl . 6 (1960), 201-203 . Depa amen de Ma emá iques Uni e si a Au ónoma de Ba celona 08193 Bella e a (Ba celona) SPAIN Rebu el 18 de Desemb e de 1991